DOI 10.1007/s00190-006-0111-3 O R I G I NA L A RT I C L E
Influence of ambiguity precision on the success rate of GNSS integer
ambiguity bootstrapping
P. J. G. Teunissen
Received: 12 March 2006 / Accepted: 14 October 2006 / Published online: 10 November 2006 © Springer-Verlag 2006
Abstract In this contribution, we study the depen-dence of the bootstrapped success rate on the precision of the GNSS carrier phase ambiguities. Integer boot-strapping is, because of its ease of computation, a pop-ular method for resolving the integer ambiguities. The method is however known to be suboptimal, because it only takes part of the information from the ambiguity variance matrix into account. This raises the question in what way the bootstrapped success rate is sensitive to changes in precision of the ambiguities. We consider two different cases. (1) The effect of improving the ambigu-ity precision, and (2) the effect of using an approximate ambiguity variance matrix. As a by-product, we also prove that integer bootstrapping is optimal within the restricted class of sequential integer estimators.
Keywords GNSS ambiguity resolution· Integer bootstrapping· Ambiguity precision
1 Introduction
Global Navigation Satellite System (GNSS) ambiguity resolution is the process of resolving the unknown cycle ambiguities of double difference (DD) carrier phase data as integers. Ambiguity resolution applies to a great variety of GNSS models that are currently in use in navi-gation, surveying, geodesy and geophysics. An overview of these models, together with their applications, can be found in textbooks such asHofmann-Wellenhof et al.
P. J. G. Teunissen (
B
)Delft Institute for Earth Observation and Space Systems (DEOS), Delft University of Technology, Kluyverweg 1, Delft 2629 HS, The Netherlands
e-mail: P.J.G.Teunissen@tudelft.nl
(1997),Leick (1995), Misra and Enge (2001), Parkin-son and Spilker (1996),Strang and Borre(1997), and Teunissen and Kleusberg(1998).
Any GNSS model can be cast in the following system of linear(ized) observation equations
y= Aa + Bb + e (1)
where y is the given GNSS data vector of order m; a and b are the unknown parameter vectors, respectively, of order n and p; and where e is the noise vector. Matri-ces A and B are assumed known. The data vector y will usually consist of the ‘observed minus computed’ sin-gle-, dual- or triple-frequency DD carrier phase and/or pseudorange (code) observations accumulated over all observation epochs. The entries of vector a are then the DD carrier phase ambiguities, expressed in units of cycles rather than range. They are known to be inte-gers, a ∈ Zn. The entries of the vector b will con-sist of the remaining unknown parameters, such as for instance baseline components (coordinates) and possi-bly atmospheric delay parameters (troposphere, iono-sphere). They are known to be real-valued, b∈ Rp.
The procedure which is usually followed for solving the linear GNSS model (1), can be divided into three steps. In the first step, one simply discards the integer constraints a ∈ Zn on the ambiguities and performs a standard least-squares adjustment. As a result, one obtains the (real-valued) estimates of a and b, together with their variance–covariance matrix
compute the corresponding integer ambiguity estimate
ˇaS= S(ˆa) (3)
with S : Rn → Zn a mapping from the n-dimensional space of real numbers to the n-dimensional space of inte-gers. Once the integer ambiguities are computed, they are used in the third and final step to correct the ‘float’ estimate of b. As a result one obtains the ambiguity resolved baseline solution
ˇbS= ˆb − QˆbˆaQ−1ˆa (ˆa − ˇaS) (4)
This solution is usually referred to as the ‘ambiguity fixed’ baseline. The quality of the estimator ˇbSdepends
on the quality of the ‘float’ solution, ˆa and ˆb, and on the quality of the integer estimatorˇaS. Different choices
of the map S : Rn → Zn, will result in different inte-ger estimators and will thus also produce differences in the probability distribution of the ‘fixed’ baseline (Teunissen 1999a).
In this contribution, we concentrate on the second step and consider the principle of integer bootstrapping. In particular, we study the dependence of its probability of correct integer estimation on the variance matrix of the ambiguity float solution. For that purpose, we first give a brief review of the theory of integer bootstrap-ping in Sect.2. We also show in this section that the bootstrapped estimator is a member from the class of sequential integer estimators. This result enables us later to determine an, albeit restricted, optimality property of integer bootstrapping, somewhat similar to the opti-mality of integer least-squares as proven inTeunissen (1999b).
In Sect.3, we study the effect of the ambiguity pre-cision on the probability of correct integer estimation from bootstrapping. Although the bootstrapped estima-tor is very easy to compute, it does not take all the infor-mation of the ambiguity variance matrix into account. This raises the question whether it takes sufficient infor-mation into account to profit from any possible precision improvement of the ambiguities. We prove that this is the case, i.e. that the probability of correct integer esti-mation of bootstrapping will always get larger when the precision of the ambiguity float solution improves.
In Sect.4, we study what happens to the performance of bootstrapping if an improper ambiguity variance matrix is used. It is shown that, with one exception, the probability of correct integer estimation always gets smaller if either a too optimistic or a too pessimistic pre-cision description is used (in the one exceptional case, the probability remains the same).
2 Integer bootstrapping 2.1 The bootstrapped estimator
Integer bootstrapping is based on the principle of sequential conditional least-squares estimation. In order to describe the process of integer bootstrapping, we start from the principle of conditional least-squares estima-tion. We have the following result from standard adjust-ment theory (Teunissen 2000).
Conditional least-squares Let the expectation and dis-persion of ˆaI = (ˆa1,. . . , ˆai−1)T ∈ Ri−1 and ˆai ∈ R be
given as E ˆaI ˆai = aI ai , D ˆaI ˆai = QI QIi QiI σi2 (5) Then the least-squares estimator of ai, when aI is
con-strained to the fixed vector zI, is given as
ˆai|I= ˆai− QiIQ−1I (ˆaI− zI) (6)
The estimatorˆai|Iis referred to as the conditional least-squares ambiguity estimator. It is conditioned on fixing the previous ambiguities to the values zj, j= 1, . . . , (i −
1). Note that ˆai|I and ˆaI are uncorrelated. This is an
important property that will be used repeatedly in the following.
The above result can be used to derive a sequential version of the conditional least-squares estimator. For i= 2, we obtain the scalar version of (6) as
ˆa2|1= ˆa2− σ21σ1−2(ˆa1− z1) (7)
in which ˆa2|1is uncorrelated withˆa1. For i= 3, the
con-ditional least-squares estimatorˆa3|2,1follows from fixing the two ambiguities a1 and a2 to the values z1 and z2.
Note, however, because ˆa3|2,1 is invariant to any regu-lar transformation of ˆa1,ˆa2, that we may as well fix ˆa1
and ˆa2|1to the values z1and z2. This has the advantage
that matrix QI of (6) becomes diagonal. As a result, we
obtain
ˆa3|2,1 = ˆa3− σ3,1σ1−2(ˆa1− z1) − σ3,2|1σ2−2|1(ˆa2|1− z2) (8)
in which ˆa3|2,1 is uncorrelated with both ˆa1and ˆa2|1. It
will be clear that we may continue in this way to obtain the corresponding expressions for the next and following ambiguities as well. The result is summarized as follows. Sequential conditional least-squares The conditional least-squares estimator ˆai|I can be computed
whereσi,j|J denotes the covariance between ˆaiand ˆaj|J,
andσj2|Jis the variance ofˆaj|J. For i= 1, ˆai|Iis set equal
toˆa1.
We are now in a position to describe the integer boot-strapping principle. In order to compute the sequential conditional least-squares solutions, one needs to specify the zjon which the conditioning takes place. In case of
bootstrapping, zj, for j= 1, . . . , n, is chosen as the
near-est integer ofˆaj|J. Hence, forˆai|Ithe conditioning takes
place on the nearest integers of all previous i− 1 condi-tional estimates. The ith component of the bootstrapped solution itself is then given as the nearest integer ofˆai|I.
We thus have the following definition.
Definition [Integer bootstrapping] Letˆa = (ˆa1,. . . , ˆan)T
∈ Rn be the ambiguity float solution and let ˇa
B
= (ˇaB,1,. . . , ˇaB,n)T ∈ Zndenote the corresponding
inte-ger bootstrapped solution. The entries of the bootstrapped ambiguity estimator are then defined as
ˇaB,1 = [ˆa1]
ˇaB,2 = [ˆa2|1] =ˆa2− σ21σ1−2(ˆa1− ˇaB,1)
..
.
ˇaB,n = [ˆan|N] =ˆan−jn=1−1σn,j|Jσj−2|J (ˆaj|J− ˇaB,j)
(10)
where ‘[.]’ denotes the operation of rounding to the near-est integer.
As this definition shows, the bootstrapped estima-tor can be seen as a generalization of the method of ‘integer rounding’. If n ambiguities are available, one starts with the first ambiguityˆa1and rounds its value to
the nearest integer. Having obtained the integer value of this first ambiguity, the real-valued estimates of all remaining ambiguities are then corrected by virtue of their correlation with the first ambiguity. Then the sec-ond, but now corrected, real-valued ambiguity estimate is rounded to its nearest integer. Having obtained the integer value of the second ambiguity, the real-valued estimates of all remaining n− 2 ambiguities are then again corrected, but now by virtue of their correlation with the second ambiguity. This process is continued until all ambiguities are accommodated. Thus the boot-strapped estimator reduces to ‘integer rounding’ in the case that correlations are absent, i.e. in case the ambi-guity variance matrix is diagonal.
Note that the bootstrapped estimator is not unique. Changing the order in which the ambiguities appear in vectorˆa will already produce a different bootstrapped estimator. Although the principle of bootstrapping remains the same, every choice of ambiguity parame-trization has its own bootstrapped estimator.
2.2 The class of sequential integer estimators
The earlier bootstrapped estimator is member of a wider class of sequential integer estimators. This class is defined as follows.
Definition [Sequential integer estimation] Let ˆa = (ˆa1,. . . , ˆan)T ∈ Rn be the ambiguity float solution.
Then ˇa = (ˇa1,. . . , ˇan)T∈ Znis a sequential integer
esti-mator ifˇai = ˆai+ i−1 j=1rij(ˆaj− ˇaj) , i= 1, . . . , n, or, in vector–matrix form, if
ˇa = [ˆa + (R − In)(ˆa − ˇa)] (11)
with R a unit lower triangular matrix and where ‘[.]’ denotes componentwise rounding to the nearest integer. We now show that the bootstrapped estimator ˇaB is
indeed a member of this class. We have the following result.
Theorem 1 Let ˆa ∈ Rn be the ambiguity float solution and let the unit lower triangular decomposition of its var-iance matrix be given as Qˆa = LDLT. The entries of L and D are then given as
(L)ij= ⎧ ⎨ ⎩ 0 for 1≤ i < j ≤ n 1 for i= j σi,j|Jσj−2|J for 1≤ j < i ≤ n and D= diag(. . . , σj2|J,. . .) (12)
and the bootstrapped estimator ˇaB ∈ Zn of (10) can be
expressed as ˇaB=
ˆa + (L−1− In)(ˆa − ˇaB)
(13) Proof From (9), it follows that the difference(ˆai− zi)
may be written in terms of the differences(ˆaj|J− zj), j =
1,. . . , i, as (ˆai−zi) = (ˆai|I−zi)+
i−1
j=1σi,j|Jσj−2|J (ˆaj|J−zj).
When written out in vector-matrix form, this gives ⎡ ⎢ ⎢ ⎢ ⎣ ˆa1− z1 ˆa2− z2 .. . ˆan− zn ⎤ ⎥ ⎥ ⎥ ⎦= ⎡ ⎢ ⎢ ⎢ ⎣ 1 l21 1 .. . ... . .. ln1ln2. . . 1 ⎤ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎣ ˆa1− z1 ˆa2|1− z2 .. . ˆan|N− zn ⎤ ⎥ ⎥ ⎥ ⎦ (14)
with lij = σi,j|Jσj−2|J , for 1≤ j < i ≤ n. Since the
sequen-tial conditional least-squares ambiguities are mutually uncorrelated, their variance matrix is diagonal. As a con-sequence the variance matrix of the ˆai is given a
trian-gular decomposition when the error propagation law is applied to (14). We therefore have the following relation betweenˆa = (ˆa1,. . . , ˆan)T,ˆac= (ˆa1,ˆa2|1,. . . , ˆan|N)Tand
the unit lower triangular decomposition of the ambiguity variance matrix:ˆa−z = L(ˆac−z) and Qˆa= LDLT. If we
rewriteˆa−z = L(ˆac−z) as ˆac= ˆa+(L−1−In)(ˆa−z) and
use z= [ˆac] = ˇaB, the resultˇaB= [ˆa+(L−1−In)(ˆa−ˇaB)]
With the bootstrapped estimator being a member of the class of sequential integer estimators, one may wonder how its performance compares with the performance of other members from this class. In Sect.4, it will be shown that the bootstrapped estimatorˇaBis the optimal
estimator of this class.
Note that the bootstrapped estimator is determined by the triangular matrix L. Hence, the estimator takes only part of the information of the ambiguity variance matrix Qˆa into account. The matrix D of conditional variances does not play a role in the bootstrapped map-ping. But as we will see in the next section, matrix D contains all the information for determining the success rate of bootstrapping.
Equation (13) provides an alternative way of comput-ing the bootstrapped estimator, one which is particularly useful when the unit upper triangular decomposition of the inverse of Qˆa is given. Let this decomposition be given as Q−1ˆa = UUT. Then Qˆa = U−T−1U−1 = LDLT. From the uniqueness of the triangular decom-position, it follows that L−1= UTand D= −1. Thus if the unit upper triangular decomposition of the inverse of the variance matrix is given, one can compute the bootstrapped estimator using UT.
2.3 The probability mass function of the ambiguity bootstrapped estimator
Let B denote the bootstrapped mapping. Then B : Rn→ Zn, because the bootstrapped estimator maps the real-valued float ambiguity vectorˆa to the integer vector ˇaB.
Since the bootstrapped estimator maps different real-valued ambiguity vectors to the same integer vector, the bootstrapped estimator is a many-to-one map. One can therefore assign a subset Bz⊂ Rnto each integer vector
z∈ Znas Bz=
x∈ Rn| z = B(x), z∈ Zn (15) The subset Bz contains all real-valued ambiguity
vec-tors that will be mapped by B to the same integer vector z ∈ Zn. This subset is referred to as the bootstrapped pull-in region of z (Jonkman 1998;Teunissen 1998). It is the region from which all ambiguity float solutions are pulled to the same fixed ambiguity vector z.
The bootstrapped pull-in regions are given as Bz= x∈Rn||cTi L−1(x−z)|≤1 2, i=1,. . ., n , ∀z∈Zn (16) where L is the unit lower triangular matrix of Qˆa = LDLTand cidenotes the ith canonical unit vector
hav-ing a 1 as its ith entry and zeros otherwise. To see this,
consider the relationˆac− z = L−1(ˆa − z). According to
(10), the integer vector z equals the bootstrapped solu-tion when rounding to the nearest integer of each of the components of ˆac− z gives zero, or similarly, when the
absolute values of these components are all less than or equal to 1/2. Since this is equivalent to stating that the absolute values of all the components of L−1(ˆa − z) are required to be less than or equal to 1/2, the result (16) follows.
The bootstrapped pull-in regions are translated cop-ies of each another(Bz = z + B0,∀z ∈ Zn) and cover
the whole space without gaps and overlaps(∪z∈ZnBz= Rn and IntBz1 ∩ IntBz2 = ∅, ∀z1, z2∈ Z
n, z
1= z2).
The bootstrapped pull-in regions can be used to deter-mine the distribution of the bootstrapped estimator. SinceˇaB= z ⇐⇒ ˆa ∈ Bzandˆa ∼ N(a, Qˆa), the
proba-bility mass function (PMF) ofˇaBis given as
P(ˇaB= z) = Bz (2π)−n2 detQ−1ˆa exp −1 2(x−a) TQ−1 ˆa (x−a) dx, ∀z∈Zn(17) It is the integral of the multivariate normal distribu-tion over the bootstrapped pull-in region Bz. As the
following theorem shows, the multivariate integral can be expressed as a product of n univariate integrals. Theorem 2 Letˆa be distributed as N(a, Qˆa), a ∈ Zn, and
letˇaBbe the corresponding integer bootstrapped
estima-tor. Then P(ˇaB= z) = n i=1 1− 2lTi (a − z) 2σˆai|I + 1+ 2lTi (a − z) 2σˆai|I − 1 , ∀z ∈ Zn (18) with (x) = x −∞ 1 √ 2π exp −1 2v 2 dv
and where li is the ith column vector of the unit upper
triangular matrix L−T andσˆa2
i|I is the variance of the ith least- squares ambiguity obtained through a conditioning on the previous I= {1, . . . , (i − 1)} ambiguities.
The proof is given in Teunissen (2001). Note that the bootstrapped PMF is symmetric about the mean of ˆa. This implies that the bootstrapped estimator ˇaB is an
of the decomposition Qˆa = LDLT are given. Finally note that the PMF reaches its maximum at its point of symmetry. Thus maxzP(ˇaB = z) = P(ˇa = a). This is
a reassuring result, since it implies that the bootstrap probability of correct integer estimation is largest of all probability masses. This probability will be referred to as the bootstrapped success rate.
3 The effect of improving the ambiguity precision The goal of ambiguity resolution is to estimate the unknown integer ambiguity vector a. We know that the integer bootstrapped estimator is unbiased, i.e. E(ˇaB) =
a. This is a nice result, since it implies that one can expect the outcome ofˇaBto be correct on the average. In order
to judge the performance of the integer estimator, how-ever, the property of unbiasedness is too weak a prop-erty to rely on. What we need is the frequency with which one can expect to obtain correct results. This frequency is provided by the probability of correct integer esti-mation, the success rate P(ˇaB = a). The bootstrapped
success rate follows from setting z= a in (18), as P(ˇaB= a) = n i=1 2 1 2σˆai|I − 1 (19) Note that it is completely driven by the sequential con-ditional variances σ2
ˆai|I, and thus by the entries of the diagonal matrix D in the triangular decomposition Qˆa= LDLT.
As was mentioned earlier, the outcome of bootstrap-ping depends on the chosen ambiguity parametrization. Bootstrapping of DD ambiguities, for instance, will pro-duce an integer solution that generally differs from the integer solution obtained from bootstrapping of repara-metrized ambiguities. Since this dependency also holds true for the bootstrapped PMF, one has some important degrees of freedom left for improving (19).
In order to improve the bootstrapped success rate, one should work with decorrelated ambiguities instead of with the original ambiguities. The method of boot-strapping performs relatively poorly, for instance, when applied to the DD ambiguities. This is due to the usu-ally high correlation between the DD ambiguities. Boot-strapping should therefore be used in combination with a volume preserving decorrelating Z-transformation. Such a transformation reduces the sequential condi-tional variances and therefore enlarges the bootstrapped success rate. Thus ifˆa is the DD float solution, a larger success rate is possible if bootstrapping is applied to ˆz = Zˆa. In case of multi-frequency GNSS, an example of such a Z-transformation is provided by the transfor-mation to widelane ambiguities. This has been
demon-strated analytically inTeunissen(1997). However, one can even do better than this by using the decorrelating Z-transformation of the LAMBDA method. This trans-formation decorrelates the ambiguities further and thereby achieves a further reduction of the values of the sequential conditional variances. For more informa-tion on the LAMBDA method, the reader is referred to Teunissen (1993, 1995) and de Jonge and Tiberius (1996a) or to the textbooks Hofmann-Wellenhof et al. (1997),Strang and Borre(1997),Teunissen and Kleus-berg (1998), Misra and Enge (2001). Practical results obtained with it and suggested improvements, can be found, for example, in Boon and Ambrosius (1997), Boon et al.(1997),Chang et al.(2005),Cox and Brading (1999),Dai et al.(2005),de Jonge and Tiberius(1996b), de Jonge et al. (1996), Han (1995), Jonkman (1998), Moenikes et al. (2005), Peng et al. (1999), Svendsen (2005), Tiberius and de Jonge (1995), Tiberius et al. (1997).
The above method of improving the success rate makes use of the lack of invariance of the bootstrapped estimator for ambiguity reparametrizations. It is not based on a change of the strength of the underlying model. It seems reasonable however to ask of an inte-ger estimator that it has the property that its success rate increases when the precision of its input gets better. This property is yet to be proven for the bootstrapped estimator.
Before stating our result, we first specify what we mean by ‘better precision’. Let Q1 and Q2be the
var-iance matrices of two float solutions ˆaQ1 and ˆaQ2. The precision ofˆaQ1 is then said to be better than the preci-sion of ˆaQ2, when the variance of every linear function of ˆaQ1 is smaller than the variance of the same func-tion of ˆaQ2. Thus fTQ1f < fTQ2f must hold for every f ∈ Rn\ {0}. This is equivalent to stating that Q2> Q1,
or that matrix Q2− Q1is positive definite. We have the
following result.
Theorem 3 Let ˆaQ ∼ N(a, Q), with a ∈ Zn, and let ˇaQ
B
be the corresponding integer bootstrapped estimator of a. Then P ˇaQ1 B = a > PˇaQ2 B = a if Q2> Q1 (20)
Proof Let the triangular decompositions of Q1and Q2
be given as Q1 = L1D1LT1 and Q2 = L2D2LT2,
respec-tively. Furthermore, define f = L−T2 g, the unit lower triangular matrix L = L−12 L1 and the canonical unit vector ci= (. . . , 0, 1, 0, . . .)Thaving a one as its ith entry
Q2 > Q1 ⇔ fTQ 2f > fTQ1f , ∀f ∈ Rn\ {0} ⇔ fTL 2D2LT2f > fTL1D1LT1f , ∀f ∈ Rn\ {0} ⇔ gTD 2g> gTLD1LTg, ∀g ∈ Rn\ {0} ⇒ cT iD2ci > cTiLD1LTci, i= 1, . . . , n ⇔ (D2)ii> (D1)ii+ i−1 j=1(L) 2 ij(D1)jj, i= 1, . . . , n ⇔ (D2)ii> (D1)ii, i= 1, . . . , n
This shows that Q2 > Q1 implies that the sequential
conditional variances of Q2 are always strictly larger
than their counterparts of Q1. This proves (20). Note
that the converse is not true. That is, P ˇaQ1
B = a > P ˇaQ2 B = a
does not imply that Q2> Q1.
The above result states that the bootstrapped success rate always gets larger when the precision of the float solution improves. Thus every precision improvement that one can realize in the underlying model (e.g. by including more data or more precise data) will directly benefit the bootstrapped ambiguity resolution. The above result also implies that in case the inverse vari-ance matrix of the observations is used as weight matrix, the least-squares method is the best method for comput-ing the float solution. Such a least-squares estimator is namely known to be a best linear unbiased estimator. It is the estimator which has the best precision of all linear unbiased estimators.
4 The effect of using an approximate ambiguity variance matrix
Apart from knowing the relation between the success rate and the actual ambiguity precision, it is also of importance to know the relation between the success rate and a presumed ambiguity precision. In other words, what happens to the success rate if the computed bootstrapped estimator is based on a too optimistic description of the ambiguity precision or on a too pessi-mistic description of the ambiguity precision? In either case, one would want the success rate not to increase. This property is yet to be proven for the bootstrapped estimator. The proof is given by the following theorem. Theorem 4 Letˆa ∼ N(a, Q), with a ∈ Zn, and letˇaQB be the corresponding integer bootstrapped estimator of a. Furthermore letˇaBbe the integer bootstrapped estimator constructed on the basis of the positive definite matrix instead of Q. Then
P ˇaB = a ≤ P ˇaQB = a (21)
with strict inequality if the unit triangular factors of and Q differ.
Proof Let the triangular decompositions of and Q be given as = LDLT and Q = LQDQLTQ,
respec-tively. Then P ˇaB = a = 1 (2π)n/2 B,a 1 (detDQ)1/2 × exp −1 2(x − a) T(L QDQLTQ)−1(x − a) dx P ˇaQB = a = 1 (2π)n/2 BQ,a 1 (detDQ)1/2 × exp −1 2(x − a) T(L QDQLTQ)−1(x − a) dx with pull-in regions
B,a=x∈ Rn | |cTiL−1 (x − a)| ≤ 1 2 , i= 1, . . . , n BQ,a= x∈ Rn| |cTi L−1Q (x − a)| ≤ 1 2 , i= 1, . . . , n Note that the above two integrals only differ in their region of integration. Using the change of variables for-mula for integrals, we now apply the transformation T : x= LQy+ a to both integrals. This gives,
P ˇaB = a = 1 (2π)n/2 T−1(B,a) 1 (detDQ)1/2 × exp −1 2y TD−1 Q y dy P ˇaQB = a = 1 (2π)n/2 T−1(BQ,a) 1 (detDQ)1/2 × exp −1 2y TD−1 Q y dy where T−1(B,a) = {y ∈ Rn| |cTiLy| ≤ 1 2 , i= 1, . . . , n} T−1(BQ,a) = {y ∈ Rn| |cTiy| ≤ 1 2 , i= 1, . . . , n} with the unit lower triangular matrix L= L−1 LQ. Note
that the two integrals have identical outcomes if L= In,
that is, if L = LQ. In all other cases, their outcomes
will differ. For these cases we now show that P ˇaB = a < P ˇaQB = a . With DQ = diag
σ2
1,σ22|1,. . . , σn2|N
and(L)ij= lij, we may write the multivariate integral of
P ˇaB = a = i1 exp −1 2 y2 1 σ2 1 σ1 √ 2π ⎛ ⎜ ⎜ ⎝. . . ⎛ ⎜ ⎜ ⎝ in−1 exp −1 2 y2 n−1 σ2 n−1|N−1 σn−1|N−1 √ 2π ⎛ ⎜ ⎜ ⎝ in exp −1 2 y2 n σ2 n|N σn|N√2π dyn ⎞ ⎟ ⎟ ⎠ dyn−1 ⎞ ⎟ ⎟ ⎠ . . . ⎞ ⎟ ⎟ ⎠ dy1
with the n intervals i1:|y1| ≤ 1/2
i2:|l21y1+ y2| ≤ 1/2
.. . ...
in:|ln1y1+ ln2y2+ · · · + ln,n−1yn−1+ yn| ≤ 1/2
For the innermost integral we have the inequality in exp −1 2 y2 n σ2 n|N σn|N √ 2π dyn < |yn|≤1/2 exp −1 2 y2 n σ2 n|N σn|N√2π dyn= % 2 % 1 2σn|N & −1 &
since the interval|yn| ≤ 1/2 is symmetric with respect to
the origin, whereas the interval in, which has the same
length as|yn| ≤ 1/2, is not symmetric with respect to the
origin. Proceeding in this fashion from the innermost integral to the outer integral, we obtain the inequality P ˇaB = a < n i=1 % 2 % 1 2σi|I & − 1 & = P ˇaQ B = a
This theorem shows that, with one exception, the use of an improper ambiguity variance matrix (too optimis-tic or too pessimisoptimis-tic) will always result in a smaller bootstrapped success rate. The exception occurs when the variance matrix Q and its approximation have the same triangular factor. This case, however, is not likely to occur in practice. The conclusion reads therefore that also in case of integer bootstrapping it directly pays off to improve upon the approximation of the underlying mathematical model. For GNSS, the functional model (observation equations) is sufficiently known and well documented. The same cannot yet be said however of the precision description of the GNSS data. Of course, a systematic study of the stochastic model is far from triv-ial. Not only do the noise characteristics depend on the mechanization of the measurement process and there-fore on the make and type of the receiver used, but the random residual terms such as environmental effects, will also have their influence. Fortunately the interest in the topic of improved stochastic modelling is gaining
ground in GNSS research and will have a positive effect on the bootstrapped success rate.
It was shown that the bootstrapped estimator is a member of the class of sequential integer estimators. With the above result, we have as a direct by-product that the bootstrapped estimator is the optimal estimator within this restricted class.
Corollary Let ˆa ∼ N(a, Q), with a ∈ Zn, and let ˇaQB be the corresponding integer bootstrapped estimator of a. Then
P ˇaQB = a ≥ P(ˇa = a)
for any sequential integer estimatorˇa = [ˆa + (R − In)(ˆa −
ˇa)], where R is a unit lower triangular matrix.
Note, when R is chosen as R = In, that the
sequen-tial integer estimator reduces toˇa = [ˆa], i.e. the integer estimator based on componentwise rounding. Hence, as another by-product, we have that P ˇaQB = a ≥ P([ˆa] = a). Thus the success rate of componentwise rounding will never be larger than the bootstrapped success rate.
5 Concluding remarks
Integer ambiguity bootstrapping is, because of its ease of computation, a popular method for resolving the integer GNSS carrier phase ambiguities. The method is, how-ever, suboptimal, since it only takes part of the informa-tion from the ambiguity variance matrix into account. It is therefore of importance to know how the success rate (i.e. the probability of correct integer estimation) of integer bootstrapping relates to the precision of the ‘float’ ambiguities. The following two cases were con-sidered in this contribution: (1) the effect of improving the ambiguity precision, and (2) the effect of using an incorrect, or approximate, ambiguity variance matrix.
that one should use the principle of least-squares for computing the ‘float’ solution.
We also investigated what happens to the success rate of integer bootstrapping, if an incorrect ambiguity var-iance matrix is used (either too optimistic or too pes-simistic) for computing the bootstrapped solution. It was shown that the success rate of integer bootstrap-ping based on an incorrect ambiguity variance matrix is always less than or equal to the success rate of integer bootstrapping based on the correct ambiguity variance matrix. There is a strict inequality between the two suc-cess rates, if the unit triangular factor of the incorrect ambiguity variance matrix differs from the unit triangu-lar factor of the correct ambiguity variance matrix.
The ease with which integer bootstrapping can be computed stems from its sequential character. That is, in contrast with, for instance, integer least-squares, no integer search needs to be performed for integer boot-strapping. We defined the class of integer sequential estimators and showed that integer bootstrapping is a member of this class. Although integer bootstrapping is suboptimal in the class of integer estimators, we have shown that it is optimal within the more restricted class of integer sequential estimators. Hence, just like integer least-squares has the largest possible success rate of all integer estimators, integer bootstrapping has the largest possible success rate of all integer sequential estimators.
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